68 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA it follows that for each point (xl9 . . . , xn) in E1 x • • • x En , we have (12.14.11.2) />'("(*!> • • • > *«)) ^ £* />i(*i) ' "AW- Conversely, it is clear that if for each seminorm p' e F' there exist semi- norms pjeTj and a real number c>0 such that (12.14.11.2) is satisfied throughout E1 x -•• x En9 then u is continuous at the point (0, . ..,0). Now let (bl9\..9 bn) be an arbitrary point of Et x • • • x En , and write u(xl9 ...,*„)- w(^i> • • • A) = H where H runs through the 2" - 1 proper subsets of {1, 2, . . . , n}9 and for each such H put yj = bj if j e H, and yj = Xj - bj if j $ H. Then it follows immediately that u is continuous at the point (bl9 ..., bn), i.e., u is continuous on E! x • • * x En . (12.14.12) On a vector space E, let F and F' be two families of seminorms. Then the topology defined by F is finer than the topology defined by F' if and only if, for each seminorm pr e F', there exists a finite family (Pth^n of semi- norms belonging to F and a real number c > 0 such that p'(x) :g c • sup /?f(x) /or all xeE. l Apply (12.14.11) to the identity map u = 1E (12.2.1). (1 2.1 4.1 3) Finally, we remark that the notions of series and convergent series are defined in a topological vector space E just as in a normed space (5.2); propositions (5.2.2), (5.2.3), (5.2.4) and the fact that the general term of a convergent Series tends to 0 remain valid without change. If E is locally convex and its topology is defined by a family (px) of seminorms, then for a series with general term xn to converge in E it is necessary that, for each s > 0 and each scalar A, there should exist an integer /i0 such that for all n ;> n0 and q > 0. If E is a Frechet space, this condition is also sufficient, as follows from Cauchy's criterion (12.9). PROBLEMS 1. Let E be a vector space over R (resp. C) and let 58 be a set of subsets of E satisfying conditions (VO and (Vn) of Section 12.3, Problem 3, together with the following: (EVi) Every set V e 95 is balanced and absorbing. • • x En , and for each j let Xj > 0 be such